Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

Categorical Sheaves

Posted by David Corfield

…there should exist an interesting notion of categorical sheaves, which are sheaves of categories rather than sheaves of vector spaces, useful for a geometric description of objects underlying elliptic cohomology.

But having listed a set of desiderata, argue that such a theory won’t be forthcoming. They resort instead to derived categorical sheaf theory.

As we will see the notion of 2-vector spaces appear naturally in this setting as the dualizable objects, exactly in the same way that the dualizable modules are the projective modules of finite rank. After arguing that this notion of 2-vector space is too rigid a notion to allow for push-forwards, we will consider dg-categories instead and show that they can be used in order to categorify homological algebra in a similar way as linear categories categorify linear algebra.

Surely patrons of the Café have something to say about this.

Posted at April 9, 2008 9:49 AM UTC

TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1653

that ω\omega-categorical sheaves, hence sheaves with values in ωCat\omega\mathrm{Cat}, hence ω\omega-categories internal to sheaves (all on your site of choice) albeit much “stricter” than other models for “∞\infty-stacks” are not only convenient and useful for describing higher bundles (principal or associated, with or without connection) hence higher cohomology, but also sufficient.

The fact that makes this work (in this context) is the central result in arXiv:0705.0452 for n=1n=1 and its generalization to n=2n=2 and higher nn refining the discussion of math.DG/0511710 which I have talked about a lot here, and which is about to be published, which says that an nn-bundle with structure nn-group GG on X, what other call a GG-torsor for the constant GG-sheaf, in terms of an nn-categorical total space fibration
P↓X
\array{
P
\\
\downarrow
\\
X
}
are equivalent to fiber-assigning functors
GTor↑X
\array{
G\mathrm{Tor}
\\
\uparrow
\\
X
}
which are all those such functors which admit a local trivialization that induces descent in the sense of Ross Street with coefficients in the ω\omega-category valued sheaf
ωCat(−,BG).
\omega\mathrm{Cat}(-,\mathbf{B}G)
\,.

While nn-categorical total space fibrations
P→X
P \to X
form an nn-stack, since their pullback is defined “only” by a universal property and hence respects composition of pullback functions only weakly, the fiber-assigning functors
X→GTor
X \to G\mathrm{Tor}
pull back simply by precomposition
Y→X→GTor
Y \to X \to G\mathrm{Tor}
and hence always form a rectifiednn-stack, in fact a sheaf.

The same game can be played with nn-vector spaces replacing GTorG\mathrm{Tor} by nVect\mathrm{n}\mathrm{Vect}.

For instance some of the aspects about K-theory and 2-vector bundles that Toën and Vezzosi mention in the introduction are described in my slides

Toën and Vezzosi cite Stolz-Teichner’s comparatively old proposal for realizing elliptic cohomology but do not cite the combined result of math.QA/0504123 and arXiv:0801.3843/ arXiv:math/0612549 which shows that Stolz-Teichner’s String bundles indeed are 2-bundles (bundles with categorical fibers, as Toën and Vezzosi are looking for).

I have talked at various times about the corresponding associated 2-vector bundles coming from the canonical 2-representation of the String 2-group, for instance in

These representations are on a sub-2-category of 2Vect:=Vect−Mod2\mathrm{Vect} := Vect-\mathrm{Mod} considerably larger than the Kapranov-Voevodsky 2-vector spaces, and I have argued many times that the fact that Baas-Dundas-Rognes didn’t find the expected elliptic cohomology from KV-2vector bundles but something else (K-theory of K-theory) is not all that surprising, given that they are working with such a small subcategory of all 2-vector spaces.

From the Witten genus argument one expects elliptic cohomology to be related to String group representations like K-theory is related to Spin group representations, so I expect that 2-vector bundles associated by the canonical 2-rep of the String 2-group to principal String 2-bundles should yield ellitpic cohomology, essentially following the original argument by Stolz-Teichner, but refining it properly 2-categorically.
I am not the topologist who could compute the corresponding spectrum. But I keep mentioning it in the hope that somebody looks into it who does.

In my notes on nonabelian differential cohomology I start discussing how the L∞L_\infty-connections and their characteristic classes of arXiv:0801.3480 may be integrated to nonabelian differential cocycles, thus yielding characteristic classes for these.

Re: Categorical Sheaves

You have all the push-forwards you want

This I didn’t say. The recent discussion on Limits and push-forward was supposed to educate me further on the question of push-forward of fiber-assigning functors (or transport functors). I wish I understood this better.

I was hoping to be able to reduce the question to some nice abstract machinery, making full use of the fact that all my nn-bundles with connection are nn-functors (ω\omega-functors).

Push-forward of these is Kan-extension. In this comment, following lots of help from Robin Houston, I recall how the formula for the Kan extension of a functor, applied to a very simple toy case, does precisely encode the idea of push-forward of vector bundles, in that the coend formula sums up all the contributions of fibers that are being sent to the same point downstairs.

So if we consider an ordinary vector bundle as a fiber-assigning functor
F:X→Vect,
F : X \to Vect
\,,
where VectVect is the category of not-neccesarily finite vector spaces, then the Kan-extension of that functor along some map X→YX \to Y produces the fiber assigning functor on YY which represents the pushed-forward vector bundle. (One needs to be careful with the extra structure around, though, like smoothness for instance).

I am trying to use the fact that Kan extensions work in the arbitrary enriched context and that I think I am able to realize nn-vector spaces entirely in terms of strict globular nn-categories to define the push-forward of my fiber-assigning (parallel transport) VV-functor
X→T
X \to T
for V=ωCatV = \omega\mathrm{Cat} to be the (left) Kan extension along the corresponding morphism. I was thinking that general existence theorems on the Kan extension when the domain VV-category is small (as it is for the cases of interest) would guarantee that this exists, and I’d just need to work it out.

Re: Categorical Sheaves

I should say what this ω\omega-categorical formulation of ω\omega-vector spaces is that I am referring to:

given an L∞L_\infty-algebra gg and a cochain complex vv (all assumed to be finite dimensional here) I am defining a representation of gg on vv to be an extension of the DGCA CE(g)CE(g) by the DGCA ∧•v*\wedge^\bullet v^*:
∧•v*←CE(g,v)←CE(g).
\wedge^\bullet v^*
\leftarrow
CE(g,v)
\leftarrow
CE(g)
\,.
This is supposed to be the dual differential version of the action groupoid
V→V//G→BG.
V \to V//G \to \mathbf{B} G
\,.
Ordinary reps of Lie algebras are of this form and every L∞L_\infty-algebra has an adjoint rep on itself with this definition. The BRST-complex is also a special case, as we discussed.

So that makes me think that this definition is “good” and “right”. But this then means that I can deduce what the good and right ω\omega-vector spaces are which we need to ω\omega-vector bundles, since they should come from reps of L∞L_\infty-algebras.

We integrate an L∞L_\infty-algebra gg by applying first the functor
S:DGCAs→SmoothSpaces
S : DGCAs \to SmoothSpaces
which is part of the adjunction induced by the ambimorphic sheaf of forms, and then form path ω\omega-groupoids, Πn\Pi_n or Πω\Pi_\omega.

The right item will integrate to B(−)\mathbf{B}(-) of the “simply connected” Lie ω\omega-group integrating gg, so this wants to become
V→V//G→BG
V \to V//G \to \mathbf{B}G
with the left ω\omega-groupoid
V=Πω(S(∧•v*))
V = \Pi_\omega(S(\wedge^\bullet v^*))
being the ω\omega-vector space that GG is represented on.

Such a VV is a bit of a mixture of pure chain complexes (“Baez-Crans ∞\infty-vector spaces”) and something richer. Its space of kk-morphisms is the collection of vv-valued differential forms on DkD^k modulo this and that and being closed or satisfying some closure relations depending on the differential structure of vv.

Then an associated ω\omega-vector bundle would be a transport functor
Πω(x)→ωCat
\Pi_\omega(x) \to
\omega\mathrm{Cat}
with local VV-structure following our general prescription, possibly with some quotients applied on the right ω\omega-groupoid if we don’t want to talk about the “simply connected” structure group GG obtained from the integration process, but some quotient by it, usually by Bkℤ\mathbf{B}^k \mathbb{Z}s.

I was thinking that by just using abstract nonsense on ωCat\omega Cat-enriched functors I get a guarantee of push-forwards of such ω\omega-vector bundles. But I still need to think much more about this.

Re: Categorical Sheaves

With regard to the T and V article which inspired this thread and especially the remark about `categorified homological algebra’, how much of the implied contortions are due to wanting to do this for algebraic geometry?

Re: Categorical Sheaves

I cannot help feeling that somewhere or other the Kan extensions being used need souping up to be ‘homotopy Kan extensions’ especially in the enriched setting. Am I missing something?

I need to have a closer look at examples. If the ordinary Kan-extension of my ω\omega-functor exists, it is hard to imagine how it would not be the right answer to some question related to push-forward. But I am really not done with thinking about this.

One aspect that I need to clarify further is that there should be a secret shift of dimension at work here.

There is this fun toy example, which should be close to your heart, because it connects this to the Yetter model and your formula for the measure on its configuration space:

Let G(2)G_{(2)} be a strict 2-group coming from some crossed module (H→tG→αAut(H))(H \stackrel{t}{\to} G \stackrel{\alpha}{\to} Aut(H)), let Σ\Sigma be some compact manifold and write Π2(Σ)\Pi_2(\Sigma) for the strict 2-groupoid obtained by picking a triangulation of Σ\Sigma with v0v_0 vertices and v1v_1 edges and some number of faces, which generate the 2-morphisms in Π2(Σ)\Pi_2(\Sigma).

Then
conf=hom2Cat(Π2(Σ),BG(2))
conf =
hom_{2Cat}(\Pi_2(\Sigma),\mathbf{B}G_{(2)})
is the “space” (2-groupoid) of fields of the Yetter model.

Let
F:conf→Cat
F : conf \to \mathrm{Cat}
be a 2-functor on confconf which is a direct sum of representable 2-functors.

Postcomposing with Tom Leinster’s category cardinality
|⋅|:(Cat,⊕)→(ℚ,+)
|\cdot| : (Cat,\oplus) \to (\mathbb{Q},+)
we can think of this as a ℚ\mathbb{Q}-valued gauge-invariant function on configuration space.

|F(−)|:conf∼→ℚ.
|F(-)| : conf_\sim \to \mathbb{Q}
\,.

I think Tom Leinster’s theorem expresses the cardinality of the left Kan-extension of FF along conf→ptconf \to \mathrm{pt} as
|∫confF|=|colimF|=∑a∈Obj(conf)|F(a)|dμ(a),
|\int^{conf} F|
=
|colim F|
=
\sum_{a \in Obj(conf)}
|F(a)| \; d\mu(a)
\,,
where
dμ:a↦|G|−v0|H|v0−v1
d\mu : a \mapsto |G|^{-v_0} |H|^{v_0-v_1}
is the measure on the configuration space which you discuss in your article with João Martins .

So this seems to be an example where we do the ordinary Kan-extension of CatCat-enriched functors and do get the right answer.

This kind of example made me start thinking that it might be indeed reasonable to just work strictly in the ω\omega-category enriched context.

Re: Categorical Sheaves

My worry is that taking Kan extensions is sensitive to the representative `spaces’ used within a homotopy type. If things are nice and `fibrant/cofibrant’ or free or something, this worry disappears I seem to remember, but otherwise, ????

Re: Categorical Sheaves

Yes. Only that Bim(Vect)Bim(Vect) will probably be too simple minded and we will eventually need to look at the classifying space of the bicategory of XX-algebras and their bimodules, where XX is something like “C*C^*” or “von Neumann”
or “algebras of local nets of free fermions on the circle” or some other flavor of algebras.

In fact, it should be a parameterized notion of algebra, where the parameter is an elliptic curve. For each elliptic curve there would be a corresponding kind of algebra (“of observables of the CFT on that curve”) and hence a corresponding classifying space for the corresponding elliptic cohomology.

I did talk to 2.5 topologists about this who eventually agreed that computing |Bimod(something)||Bimod(something)| should be done. I haven’t heard back of further progress along this line for a while, though.

Meanwhile, I am wondering if we need to transfer the problem from ∞\infty-categories to topological spaces.

What I’d maybe rather want to do is understand elliptic cohomology not in terms of spectra, but in term of ∞\infty-categorical cohomology.